Bernhard Riemann: The Habilitation Dissertation

[@Kimmath1]

I'm sincerely and tremendously grateful to the devotion you put on this clip, especially the graphic explaining curvatures. I've never ever seen or heard or read any explanation as lucid as your clip. My words can describe nothing near how much I appreciate this video. A million thanks from me!

[@jasonaross]

Wow, Kim, I'm honored by your response! This was definitely a labor of love -- this paper by Riemann is one of the most important pieces of writing of all time, and it crosses between science, mathematics, and philosophy.

[@xiongjun6743]

Mr. Ross, would possible to have subtitle in this video? I understand most of it but at some key points reading English is much clearer than hearing it. I think this would be true for others whose first language is not English.

[@jasonaross]

Thank you for the comment - yes, I will add the subtitles soon!

[@jehovajah]

Euclid is often sidelined by modern thinkers as not doing x yz. This is usually to promote the modern thinker, not the pragmatic nature of Euclid's teaching material. Greek philosophers from Pythagoras have well observed these relations between the subjective and objective relations of form. Euclid does not avoid these discussions, he pragmatises them. To do this he uses the notion of monas. Euclid therefore teaches a general method of apprehending space.

[@douglasstrother6584]

Good ol' High School Geometry was the thing that turned-on my Math-lightbulb. What one can construct (a proof, furniture, a cathedral, etc.) with only a straightedge & compass is astounding! Upon that foundation, one can truly appreciate Riemann's vision.

[@samisiddiqi5411]

@@douglasstrother6584 precisely this

[@douglasstrother6584]

@@samisiddiqi5411 It is also fascinating that conic sections are orbits for 1/r^2 force situations, like gravity and electrostatics.

[@j.vonhogen9650]

Jason Ross, if you read this, thank you for the excellent introduction to Riemann's Habilitations Schrift. You are really talented as a teacher! Riemann was a true genius and more people outside the world of math and mathematicians should know his name.

[@ayushthada9544]

I am 10 years late for this video, but I'm glad that I watched it.

[@krishnanchandran5175]

Geometry is not the stage in which events unfold, it is the shape of the action itself.

[@ramontravis5191]

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[@killianjayden3949]

@Ramon Travis instablaster =)

[@ramontravis5191]

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[@Schufti17]

is this a quote? if yes from whom? thanks!

[@gurushabu]

I am delighted! I read Riemann when I was a teen. He blew up my mind at that time and open the doors of a much beautiful and complex world. With this presentation you put everything in such a crystal clear view that shows how profoundly you have dived and immersed yourself on the possibilities of the theory. I think even Riemann would be very pleased to watch the superb presentation you've made. My complements!

[@jasonaross]

Thank you for your inspiring words!

[@acadianalien]

When I was introduced to the Gauss-Bonnet theorem, it blew my mind how elegant it was. I had done Topology in my Real Analysis class but its true beauty was first revealed to me in that theorem.

[@Spacexioms]

His dissertation had a ver deep philosophical impact, thanks for the insights

[@douglasstrother6584]

Good ol' High School Geometry was the thing that turned-on my Math-lightbulb. What one can construct (a proof, furniture, a cathedral, etc.) with only a straightedge & compass is astounding! Upon that foundation, one can truly appreciate Riemann's vision.

[@PrometheanAction]

Hi, thanks, if not already signed up, you can sign up to participate live in our Zoom calls and get LaRouche PAC updates at lpac.co/w2d

[@pinkunath09]

Mind blowing explanations. I read about Reiman many times, but I never understood him. I want to give millions of like and thank you for the video.

[@nal8503]

+Pinku Nath I agree, the explanations are great. As a mathematician there was little new for me except for the historical context, but I can totally see non-mathematicians becoming enthralled by presentations like these as well, which makes me excited for the future of mankind. Can't wait until the spread of knowledge and creativity becomes the next popular thing for the masses. The day this happens we'll be taking the first step into a real utopia.

[@jasonaross]

+Pinku Nath Thank you very much!

[@boobah1067]

Excellent. If you added search term Riemann (without Bernhard), you would get more views. And you could add links to the new Riemann videos and pages in the intro comments below the video or as a click option near the end or beginning, while watching the video.

[@davidzurbaran2280]

This is an understanding beyond the power of the creative mind.

[@ivanvukovic5770]

Now I gained a thorough shakedown of my views on space and forces within it. Thank you very much, very useful for people getting into 3D programs, these distinctions are something I needed to have someone make in my head. I would love to see more videos as thorough and as understandable and well explained by graphics as this one about the beauty of math, especially one about L-systems and their syntax. You sir made my day!

[@jasonaross]

Thanks, Убах! In the 18th and 19th centuries, it was common to schools to develop physical geometry kits with models and shapes to help make concepts clearer. Now 3d software makes it a lot easier. These concepts shouldn't be known only to mathematicians and advanced geometers, but as formulas they often stay that way. This is bringing the concepts to life.

[@christophervillarroelpizar4750]

This is by far the best video I've seen in KZbin. When is the second part coming to my brain?

[@ChadMello]

Judging by many of the comments here, it seems that students were directed to watch this as part of some class homework or grade ;) Personally, I think the topic, concepts, and content here are good. Computer scientists have recently began studying Riemannian manifolds with respect to their application towards improving the performance of machine learning for certain problems - particularly against positive definite matrices. Many students struggle to understand why this approach makes sense, despite having an understanding in the rudimentary mathematics behind it. I would recommend this video to anyone who desires to know "why" we might want to map objects from one space onto another or "why" we might be interested in working within bounded localities on a given n-dimensional manifold.

[@carlchen9935]

The best presentation about Riemann manifold I've ever seen.

[@sebastiangil2680]

Masterpiece of an exposition.

[@carlkuss]

This is excellent and has helped me center my ideas regarding Riemann, something that is important to me now. The only one little discerning comment that I would like to make is this. Although I am in complete agreement that Riemannian thought has had and will have repercussions over science in the whole range and depth of that term, and although this video is right in insisting that the Riemannian framework places man within the world as active and transformative and that this has implications for the field of economics and ought to transform the conception of economics humanizing it and making them more ecological, I don´t think Riemann would have seen economy as ultimate reality. He was also a believing Christian. He believed in metaphysics and in transcendent causes, as you point out in associating him with Kepler. For Riemann the physical explanation of mathematics really is an explanation. This makes sense because Riemann is not a materialist and by the physical he does not simply mean the material. With nature he is referring to the order of being. He is like the Greeks in this. If economy is everything you don´t have that transcendence. I suppose it is not the word economy that bothers me here. One also could put it this way: concluding with Riemann´s importance for economy is a move that requires more justification. One is however free to close with that focus. It is an interesting focus, in fact.

[@crazyoldhippieguy]

12-June-2020.That was intense.

[@boobah1067]

Jason, your work is so inspiring. Are you going to do a video on Gauss' quadratic reciprocity (or a couple of videos)? The pages on the subject at the Larouche site are a bit obtuse at times, for me: they cover things in detail and then sometimes make a claim without spending time on it. In particular, with so many numbers being cited, it would be nice to have more graphic comparisons: e.g. out of many: show 8n +1 (etc.) with blocks and colours. Thanks.

[@ghiribizzi]

Gauss never dare to publish something beyond Euclidian geometry because of Kant, but Riemann.......this has a profound philosophical resonation something that Einstein saw clearly

[@ΖήνωνΕλεάτης-δ7κ]

(2:05) Are you saying that, arithmetic space is non-linear? That the equation 1+1=2 does NOT imply that 1000 + 1000 =2000 ? That pi is not a constant, but it depends on the absolute diameter of the circle?

[@jasonaross]

Hi there - I was saying that *physical space cannot be assumed to be linear. Basic arithmetic such as 1+1=2 or 1000+1000=2000 can exist in an abstract world that can be considered flat. For the circle, if one means a real physical circle made in real physical space, then the answer is yes: the ratio of the circumference to the diameter can change.

[@jasonaross]

This would occur, for example, if there were a star at the center of the circle.

[@trukkstop1]

Great vid, but I must nitpick: at 6-minutes, you mention that N-parameters are needed to locate an N-dim point. This is only true if there is a "continuity" between the parameters and the spatial structure. Set theory tells us that N-space can be parameterized by a single real parameter. I mention this because the discussion is meant to establish the most fundamental properties of space, so we must keep this in mind.

[@jasonaross]

If I understand you correctly, you're pointing out that dimensionality only exists where there is continuity. I agree with you. The examples I chose were continuous ones. For a discrete multidimensional manifold (such as all points in space with integer components), the points can be counted as if they were on a single dimension. There is more on this in my video on Cantor: kzbin.info/www/bejne/aYDWaatsfJJ_g7c

[@mariodelgadillo4570]

Great presentation. Thanks

[@some1rational]

This was truly enlightening for me as a mathematics enthusiast, thank you.

[@PrometheanAction]

You're very welcome! - Jason

[@jamesfullwood7788]

Amazing lecture Jason. Thank you so much for this. The last few minutes made me think about things I have never thought before. I'm currently teaching a course on differential geometry and I will suggest all my students to watch this video as well.

[@jasonaross]

James Fullwood Thanks, James, glad to hear it. I'd be very happy to hear how it goes with your students. What struck you in the last minutes of the video?

[@jamesfullwood7788]

At 35:02. "Geometry is not the stage in which events unfold, it is the shape of the action itself." This is great way to encapsulate the philosophy of general relativity. And from 35:42 until close. "A true riemannian geometry...." I completely agree that for science to elevate to the next level it must incorporate the role of the mind and consciousness into it's scheme. Consciousness I believe is the true frontier for science (it's amusing to come across "prominent" philosophers such as Daniel Dennet who claim they've got consciousness all figured out).

[@paul1964uk]

Thank you. If I've understood the central thesis of Riemann it's that 'geometry' (by which I think is meant differential geometry) must be approached as a physical theory rather than as a logical, deductive study.

[@alwaysdisputin9930]

24:38 "But we don't see such stars or such hallows in astronomy." ⦵ Around black holes we do eg the movie Interstellar BH have accretion disks similar to Saturn's rings. The top of this ring ⦵ is from *behind* the black hole. Light from the disk that is *behind* the black hole got bent over the top of the black hole like when I brush my hair over my bald spot The bottom of the ring ⦵ is also from behind the black hole.

[@opheliaslastsurf]

At 24:20 is that not the description of gravitational lensing? We do see double stars and galaxies and objects stretched out into halos from that severe curvature caused by the gravitational force of dense galaxy clusters! The rest of the video is great though, I've come back to watch this a few times and compare the progression of my notes

[@douglasstrother6584]

Local curvature is flat for practical purposes, so Euclid's axioms have a strong self-evidence.

[@rkpetry]

[16:09] 250,000 stadia Earth circumference is peculiar to Eratosthenes, for half a century before him Aristotle recorded that the mathematicians of his day said the circumference of the Earth was 400,000, stadia, and, it is this discrepancy that has prevented searchers from finding the Isle of Atlantis that sank in a day and a night ca 2345 BCE exactly where but-only-correctably-when Plato his teacher had cited it....

[@KaderWadan]

As a physicist, I have my doubts about the so central role of space time geometry in nature and economics, because the dynamics of all the variables and parameters, governing their underlying processes, are so huge, that they can not be approximated by the riemannian or semi-riemannian geometry in any dimension. Riemannian geometry is an excellent tool for putting artificially physical, biological or economical laws in this geometry. So, the geometry can not tell us about the laws of nature. We can not deduce dynamical laws of nature and sociaty from any geometry.

[@PrometheanAction]

Exactly, you cannot deduce dynamical laws of nature from geometry. Riemann's point is that you must discover the interaction space of real processes by starting from physics, not mathematics. In this way Riemann is explicitly (and Gauss implicitly) ANTI-Euclidean, unlike the non-Euclidean work of Bolyai or Lobachevsky, who replaced the parallel postulate, but still had a geometry based on... geometry, rather than an expanding domain of discovered physical principles.

[@angelmendez-rivera351]

LaRouchePAC Videos I think your conclusion is an understatement. Mathematics and physics, or more generally, science, are fundamentally different epistemological disciplines. Mathematics are a language of logic; they are founded on deductive reasoning, axioms, and premise to conclusion inferences. Science, on the other hand, is inductive, empirical, and statistical. Science does not actually establish truth, and it does not care much about truth, because truth is a concept that we borrow from logic without much rigor or definition to do so. In mathematics, we establish truth axiomatically. We can get the principal square root of a negative number because we created an axiom that is consistent with the rest which allows us to. We can extend fields to wheels and formulate the algebra so that we can meaningfully divide by zero just because we can and we want to. Circles have zero eccentricity because we define them this way, not because we observe it. Real circles of zero eccentricity have never been observed in nature, perfect shaped as idealized in geometry are not real, but this does not change the truth that mathematics give us. On the other hand, science is not about truth. Science is about empirical adequacy. We observe a phenomenon, we want to know what this phenomenon is and why it happens, and of course we cannot use logic to simply explain the phenomenon because logic is inherently non-physical. Logic is in the realm of the abstract. Science gives us knowledge about the physical world because it was designed physically, to be physical. We make a hypothesis about what we observe, then we make an experiment and extract data. Does the data match the predictions of the hypothesis? Can this hypothesis explain already existing ideas? Then we go with it until we find something that works better statistically and in explanation. But this is not what truth is, this is far from how logic is done. And that is fine.

[@o2807]

@@angelmendez-rivera351 l vote yrs: Best comment! I found myself taken in at first

[@samisiddiqi5411]

Fantastic video. I would love to see more like it.

[@jurgenvanlunenburg9410]

I'd like to add to the concept of manifolds that some magnitudes exist only in dimensions of a certain size (e.g. taste is five dimensional since we have 5 taste types) but others exist in all dimensions (e.g. length has meaning in 1D, 2D,...,ND). Also magnitudes can have equal meaning in some dimensions and differ in others (e.g. length = position in 1D, but not in 2D).

[@LaroucheScience]

Length is position in one dimension if you can only go one way. That is, if I start in the middle of the ruler, and say "go ten centimeters away" there are two places that correspond to that. (A "circle" in one dimension is just two points.) Yes, there are distances in one, two, three, and higher dimensions, but distance itself is a simply extended manifold, since all possible _distances_ are only one dimensional. I can use a ruler to measure the distance between two points on a piece of paper, or between two locations in space, but the ruler length is still only one dimension.

[@jurgenvanlunenburg9410]

True, but the point was there is no dimension > 0 where "length" has no meaning. Whereas some only have meaning in spaces of certain dimensionalities.

[@LaroucheScience]

Jurgen van lunenburg Yes, this is true. When Riemann talks about decomposing manifolds into the component dimensions (modes of determination), he only relies on a form of "distance" to do it. Although his language may be difficult to understand, he could break up a surface by describing any point on it with two parameters: the distance from an initial point and the direction you set out in to reach it -- a sort of general polar notation. The same can be done with volumes -- how far away is the point, and which direction (2d) is it in.

[@NocturnalJin]

Seems like an orange that got bigger on the inside would translate into common experience as a change in density. I guess not the same kind of density you get by adding more matter into the same volume. More like the way space curves in the presence of mass, so more like the TARDIS I suppose. I guess from the perspective on the inside, it actually gets less dense.

[@doubop]

Brilliant! thank you! i have learn so much!

[@MS-cj8uw]

What a marvelous...thank you ...

[@JessicaAngel]

Stunning!

[@BehnamEsmayli]

+Greg Schmit Yes. Those " a line is the shortest distance between two points" in any space, and mathematical terminology for a "line" in this sense, which is no more the same as the edge of a carpenter's ruler, is "A Geodesic." On non-pathological spaces they always exist, at least locally. But be careful! Many of our intuition about flat euclidean space is not true anymore. Take north pole and south pole on a earth. Is there only ONE shortest path between them? No. Any meridian is "a" shortest path. This does not hold in plane, where there is only one straight line joining any given two points. But locally things are better. If two points are closer that north pole-south pole distance, then there will be just one shortest meridian joining them.

[@sarthakmishra6915]

This is the best video ever made on this topic ,jason you must be a great teacher

[@jasonaross]

Thank you! Glad you found it helpful!

[@sarthakmishra6915]

@@jasonaross I am glad I found your channel

[@ahmedgaafar5369]

thank you sir, this is a gorgeous lecture

[@jasonaross]

Thank you very much!

[@SuperFinGuy]

Great video, it is awesome to see a truly informative and rational series on Riemann and physics. In the same way that a curvature of a 2d space (a plane or an area) is a change in the third dimension. The curvature of a 3d space (volume without boundary) is a change in the fourth dimension (density) or in space/time. Gravity is a blatant example of a space curvature.

[@GuillermoValleCosmos]

I'm not sure I understand the quote at 25:56 I mean I don't know much about set theory or diff geometry, but I think that the metric relations are about the magnitudes and these represent the ordering of the elements on a set. I understand that in the continuous case you have an infinite number of elements. Is this the reason for what he says? Is it that you can't order such an infinitelly big set? hm I'm gonna watch your vid on cantor now..in any case thank you for this videos, they're awesome!

[@jasonaross]

Hi Guillermo -- thanks very much! Both discrete and continuous manifolds can have infinite numbers of elements / positions / points, although the kind of "infinite" is different. You're right that Cantor is the way to understand more about this.

[@albertkundrat9227]

Suppose, instead, We start with not 2 parallel lines in space attempting to extend indefinitely, but two curved lines that aren't parallel! Then the Question can be ASKED: will these 2 curved unparallel lines stay forever unparallel as they infinitely extend! Thus the curvature of space counter-curves those finitely curved line segments into infinitely extending parallel lines?

[@jasonaross]

Yes, that's another idea for an animated example. Two curves whose appearance in a certain region would lead us to believe they'd intersect, might not, if the space were negatively curved. Check out images or videos of negatively curved spaces -- in them, there can be multiple parallel lines passing through a point not on a given line (parallel in the sense that they'd never intersect the first line). The real fun with Riemann's work is his consideration that space might not have a constant curvature, but may be curved differently in different places, based on the physical principles at work. Einstein's general relativity is an example of such a non-constant curvature (of space-time, not just space).

[@albertkundrat9227]

Thank you for your encouraging reply!

[@esakkithirugnanam6626]

Could you send me this lecture in pdf format. I couldn't follow your English.

[@rkpetry]

[32:13] Einstein, no, Though his 3-point "Twins Paradox" has withstood common scrutiny his 4-point 'Acausality Paradox' exposes his "D" grade in mathematics by the simplest of Newtonian principles, integration to get there from here, where even Newton sh/c/would have calculated 'the other end is running late' and so pre-corrected Albert E. (SEE mine.)

[@davidcooper2589]

einstein didnt suck at math. he flunked that one entrance exam, but he did fine on the math section, but he was just bored to death of the other stuff

[@jryanconnelly]

"Metrical properties are then to be regarded no longer as characteristics of the geometrical figures per se, but as their relations to a fundamental configuration, the imaginary circle at infinity common to all spheres." -Felix Klein

[@jehovajah]

I thank you for your exposition of Riemann. I am intending to read his gathered works as time permits. However, you do not reveal a new philosophy, but a modern slant on a pre-socratic one. This is not to belittle modern thought, but rather to remove modern prejudice against the ancient concerns and discussions, In every way they were as coherent with modern conclusions as the term allows.

[@matematizare]

Great video. Excellent ...

[@lorddiaboboss]

Thank you very very much!

[@jasonaross]

You're very welcome, Kevin! I'm very glad to make this beautiful and important work by Riemann more broadly known! What brought you to this video?

[@boobah1067]

Hey Jason, I still hope you will get to Gauss in extensive videos as you've done for Riemann/ u speak so well. -- Note, too, pls tell Martinson: glitches in the Gauss pages: some anims become text in [ ] brackets, for example at: science.larouchepac. com/gauss/ceres/InterimII/Arithmetic/Reciprocity/Reciprocity. html , so they don't run or show. Also, there r some editing errors (not typos; I mean skipped arguments for the reader), so some logic is hard to follow. But you people are so great ...

[@jehovajah]

What may be obvious to a person is subjective. I for example think that it is obvious that the approach leads naturally to an investigation of these field effects. The terminology clearly did not exist, but electron and magnes did. The only real difference i allow is electronic devices of the complexity we are familiar with. To call Euclid a faker is of course provocative.

[@alwaysdisputin9930]

24:27 _"the same star could appear on multiple different paths appearing as a hallow rather than a point. But we don't see such stars or such hallows in astronomy."_ ⦵ Around black holes we do eg the movie Interstellar Black holes have accretion disks similar to Saturn's rings. The top of this ring ⦵ is from behind the black hole. Light from the disk that is behind the black hole got bent over the top of the black hole like when I brush my hair over my bald spot The bottom of the ring ⦵ is also from behind the black hole. (When I editted this comment, it got automatically deleted due to youtube being a buggy mess. I would've had to type it in all over again but instead I googled 'view my youtube comments' & managed to find a record of all my comments. & lo there it was. Some say black holes retain all the information that ever fell into it. Paul Sutter said it becomes trapped between 2 forces: gravity & the antigravity caused by it's immense rotation. Spinning things tend to throw stuff outwards, y'know)

[@zbjet1]

amazing video, great job

[@intensen3x]

after watching and getting slightly lost cuz me was tired...my question is -> how and why the Riemann conclude that working with just imagination to understand or possibly produce absolute geometry to fit all dimensions is not enough or wrong method and therefore should be looked into physics for answer?

[@PrometheanAction]

In a nutshell, Riemann shows that just as there are curved surfaces in which the flat space of Euclid is not valid (such as on a sphere, on which you cannot draw parallel *lines, where a line is a straight/shortest-line, like a string on an orange), there are also curved spaces. This is famously seen today in Einstein's general relativity, in which space takes on a curvature. How could we choose among the innumerable possible shapes of space? What makes it behave one way rather than another? That's fundamentally a question for physics, not mathematicians, since you can't just sit around in a room and imagine how nature works without doing experiments and learning things. Those discovered principles-and that discovery process-is the essential "shape" of real physical space in the universe.

[@gpw_bernhardb.rementilla6065]

Dr. Riemann not only places Mind@the center of science, but he also puts BOTH the Mind&Heart@the center of mathematical creation of the Universe, equally seen thru physical configuration design of the natural world by his naked eyes of mental science; & most deeply felt&further seen thru abstract configuration design of natural world by his naked eyes of mental mathematics. To him, both science & mathematics are inseparable attribute of duality principle of GOD's creation! - Bernard Rementilla.

[@rkpetry]

[30:37] that diagram of light rays refracting into a different medium is meta-messed-up...

[@scottthompson7672]

Bernard Riemman gave his father the choice of continuing in theology with the intensive study of the philology of languages or mathematics. Riemannn's father was a theologian. The biggest problem with this group is that they have not studied the very metaphysical wellsprings of the deeply religious ideas of those great scientists, they have "detailed" knowledge of. Ye, though I have all knowledge, yet do not have charity (Caritas or Agape), it is for not. Riemann's theogical studies brought him in-depth studies of the science of such as the Tilak's Artic Hymn of the Vedas in Sanskrit. I have found the lack love (Agape) in many scientists today for mankind to be among sciences greatest problems today as my relative Princess Diana pointed out. It was also the finding of Albert Einstein, who noted that all major colleges for the study of science had once been weded with divinity schools. The awe and wonder of coming to know Gods thoughts as what Nikolaus von Kues called Imago Viva Dei ("In the Image of a Living God") to be lacking often in this group's work, and Cardinal von Cusa, knew Platonic science and carried it further while being congruent with theology to help spawn a Renaissance--e.g. Leonardo da Vinci. Kepler, Leibniz all the great minds presented here were metaphyscians as well as leaders in scientific progress. However, many of the examples presented here could actually be used to develop geometry classes for pre-school students say ages 3-6.

[@qubit966]

Scott Thompson They have tons of work on Cusa. Even published translations of his works. mobile.schillerinstitute.org/site/lar_related/2013/hzl-cec-cusa.html

[@henrymeyer5214]

2:21 not hypotheses, but rather axioms

[@InfiniteUniverse88]

Is Riemann's Habilitation dissertation the basis for the claim that a universal constructor is better than a Turing machine?

[@SuperFinGuy]

@laroucheyouth Who's replying? I hope is Leandra Bernstein LOL Well since Kepler showed that motion is ontologically primal, or how I see it that objects in existence at all levels (at all scales) are primarily formed by motion. One could say that the 4th dimension is the density of action or energy of our 3d objects? Anyways, looking forward to the next installment.

[@angelmendez-rivera351]

SuperFinGuy We already know what the fourth dimension is, though. It is called time.

[@123must]

A lot of thanks !

[@realcygnus]

superb content !

[@crittinger]

I couldn't understand the animosity towards Euclid until about the end. I enjoyed the information, but I would have like a more sober presentation. I think 'wrong' is too strong of a word (with too little reverence) to describe Euclid's geometry. I thank you nonetheless

[@annalisetrite7281]

Same here, I enjoyed the video but Euclid's geometric arguments are logically valid. As they describe the physical world, they may not be sound. But they are still useful and interesting.

[@PrometheanAction]

Annalise Trite It's not that Euclid's geometric conclusions are wrong (although in the sense that they assume a flat space, they are): the more important error is that logic is not science. That is, true scientific discoveries *overthrow the axioms underlying a logical system; they are not novel or clever theorems derivable from those axioms. So Euclid took discovery and science out of geometry, and given how geometry shapes thinking of science in general, it had a profound influence. That's the problem with Euclid (and logic, in general). For more, see my article on this and Gödel: "The Failures (and Evil) of Logic: A Particularly Evil Aspect of Bertrand Russell" available in the description of this video: LPAC Weekly Report: The New Paradigm for Mankind - Jason

[@jasonaross]

The point of Riemann's habilitation lecture was to put the focus on the axioms that lie at the basis of geometry. Euclid began from axioms and postulates (including the parallel postulate) and arrived at conclusions from them, but this is not a general model for discovery. That is, logic can never address the truth or falsehood of the axioms that form its basis (other than finding a paradox among them if they interfere). I agree that there are interesting proofs among his work.

[@PoloMaldonadoM]

Well, Euclid axioms and theorems were new at the time when they were discovered/developed/created. Logic is not "evil", it's simply set theory applied to thought. Cantor made it in order to understand infinity. Euclidean geometry still works even if you call it "fake."

[@schmitgreg]

I still think that a line is the shortest distance between two points. Even if the points lie on a sphere, there is a (non-curved) line that connects them that goes through the sphere, even is it is blocked by matter (like the Earth). On second thought, maybe the shortest distance would be curved on a purely 2-dimensional sphere where the space inside the sphere doesn't exist. Hard to imagine that physical area not existing, but I think that's because I happen to live inside a 3-dimensional space, making that conception difficult. Either way, this was a very interesting video!

[@austensatterlee8139]

Yeah I think you'd need to assume a space that only exists on the surface of the sphere. It is weird though, because the notion of a sphere demands that 3rd dimension. The idea would probably work just the same if we were talking about a sphere of n-dimensions, though. I guess it's probably related to the idea that gravity could bend space. In order for our perceived 3-dimensional space to bend, wouldn't there have to be some 4th spatial dimension for it to bend into? If not, then I think the space would have to have been destroyed.

[@PrometheanAction]

Hi Greg -- yup, you have to imagine the space inside the sphere not existing, and have only the surface. Then the shortest line along the sphere is the string pulled taut along its surface. It's very hard to imagine curved spaces without thinking of them in three dimensions -- that's why I included the "flatland" people, to give an example of the paradoxes they could find, that would lead them to conclude that their "space" (surface) is curved, even if they can't think in three dimensions. That's what we do when we have curved space-time in our three dimensions of space. We can't actually imagine the four dimensions of Einsteinian space-time, but it is still possible to understand. Or when trying to imagine the curved "space" around our sun when light is bent gravitationally -- that's difficult! What does a curved space look like?

[@PrometheanAction]

***** Hi Austen. Although at first it seems impossible, it *is possible to imagine the characteristics of the surface of the sphere, or at least the characteristics of motion on it, without there being three dimensions. That's what the "Flatland" example was meant to go through -- an intrinsic concept of curvature. Riemann writes that beyond three dimensions, we can only express curved manifolds with equations, rather than with the full clothing of geometry. Yet we can still understand and work with them. So it would be the same for scientists in Flatland, who might conclude that they live on a sphere (or whatever name they'd give it), based on mathematics, even if they couldn't fully picture it, since they can't visualize three dimensions.

[@BehnamEsmayli]

+Greg Schmit Yes. Those " a line is the shortest distance between two points" in any space, and mathematical terminology for a "line" in this sense, which is no more the same as the edge of a carpenter's ruler, is "A Geodesic." On non-pathological spaces they always exist, at least locally. But be careful! Many of our intuition about flat euclidean space is not true anymore. Take north pole and south pole on a earth. Is there only ONE shortest path between them? No. Any meridian is "a" shortest path. This does not hold in plane, where there is only one straight line joining any given two points. But locally things are better. If two points are closer that north pole-south pole distance, then there will be just one shortest meridian joining them.

[@BehnamEsmayli]

+Greg Schmit As for "the space inside the sphere doesn't exist" part, try to empathize with those flat-landers imprisoned within the thin shell of the sphere. It IS indeed impossible for them to imagine an emptiness, a hole right at the heart of their world/space. Then upgrade to 3-4 dimension analogue which is us. We could just as possibly be living on a "thin" shell in 4-dimensions without having the physical abilities to detect the 4th dimension.

[@CarterColeisInfamous]

wow

[@1974JTWright]

very nice

[@jasonaross]

+john wright Thanks!

[@MarkDachilleWildhood]

This is a mind trip!

[@PrometheanAction]

Annie Dachille Thank you. It was a blast to make this video, and I think the content is absolutely essential for everyone to know!

[@PrometheanAction]

Mark Dachille Wildhood Glad you liked it!

[@MaximusAurelius]

Excellent video professor.

[@jasonaross]

+Edgar Espinoza Glad you liked it! The inside history of the development of science needs to be the basis of science education, rather than only the conclusions. Riemann is an overlooked genius!

[@123must]

Thanks a lot !

[@reimannx33]

The guy looks - or , the topology of his face is homeomorpic to Chandler's from 'Friends' LOL

[@SandeepPulla]

😂

[@quagapp]

This is very good at the start but it accelerates into a lot of philosophy that is fraught -some interesting ideas but there are whole realms of ideas he is trying to kind of "summarise". Otherwise it is very interesting and mostly well presented. Keppler and all the others could be dealt at greater length - the move from mathematics - to reality ! - to economics is admirable but it is too big a leap...some dubious conclusions, some contradictory positions also.

[@ecoguy3081]

did he say EUCLID a faker at 2.30 ?

[@Kabamaru22]

Are you referring to Euclid as a faker? 2:30

[@PrometheanAction]

Yes. Two reasons: 1. Euclid is wrong about space being flat. 2. (more important) he gives an idea of *proof and discovery that is limited by the axioms assumed to bound reality. True science is the overturning of axioms, not finding novel theorems that can be proven by existing ones. How were the geometric theorems that he describes originally discovered? How will we make *new discoveries? Not by Euclid's method. This is also the essence of Gödel, who attacked Russell's silly attempts to do to all of mathematics what Euclid had done to geometry.

[@Doutsoldome]

That's beyond ridiculous. By this line of reasoning, Newton is a faker, because he was wrong in not jumping straight to general relativity. His theory of gravitation was so flawed, so naive, so lacking in imagination, wasn't it? And wrong! Oh, what an impostor, what a faker! Also, mathematics doesn't make empirical claims about reality. It is a formal science that deals with abstract systems, like logic. Good science doesn't accept dogmas and grants that any idea can be challenged. But it also adheres to reason and mathematics can and does achieve results that we know to be true with absolute certainty. For example, there is an infinite number of prime numbers. Done. And done correctly by Euclid, by the way. Nothing to revise here, ever; move on. Saying that "true science is the overturning of axioms, not finding novel theorems that can be proven by existing ones" is misleading, because it tries to establish an apparent opposition between the skeptic nature of science in general and the cumulative nature of mathematics in particular, while such thing clearly doesn't exist. Skepticism, in mathematics, is related to the reliability of proofs. Is Euclid's proof that there is an infinite number of prime numbers correct? If you doubt it, you can check it yourself. In this case, the answer is: yes, he was correct. In addition to that, I should say that mathematics grows by both the creation (discovery?) of new structures and by proving new theorems with the help of older ones. Again, there is no contradiction. This comment is absolutely enough for me to know that you, sir, know nothing about how science really works. There is indeed one faker here, and it is not Euclid.

[@jasonaross]

See point 2 -- presenting deduction as science omits the development of new principles themselves.

[@jamescook5617]

Personally, I take a rather different view on geometry. To my mind, there are infinitely many geometries which are likely divorced from physical reality. The triumph of 19-th century math was to eventually sever the inane conflation of geometry and physics. Yes, they can be correlated, I like the dream of string theory as much as the next guy, but, there is no logical need. We are indeed free to imagine geometry without physical limitations. I am glad that (at least in real mathematical circles) there is no expectation that math be physical. We ought not conflate math and physics as it inexorably leads to illogical mysticism.

[@LudwigSauerteig]

riemann geometrie isen't mathematic it is physic in fact it is a distort, twist one because the conzept of time is a stranger to mathematic and a point isen't physical. riemannsart is to use time as not physical and a point physical. by the way two parallel never meet in eternety starting with a Algebrapoint change into even a so called astigma using the imaging of Leibniz stabelstick for both lines to Protonen and to hold the distance. No it would'nt will cosstyle.

[@jasonaross]

+James Cook It seems that this triumph required the notion that multiple geometries could exist, which Riemann spelled out in his habilitation dissertation. For Kant, Euclid, etc., there was only one possible geometry, which existed both in thought and as the physical geometry of space. The other key concept from Riemann is that the *basis of the geometry that actually corresponds to real physical space is not itself geometric: it is physical. Einstein is an excellent example of the application of this outlook, whereby physical principles shape the requirements of space, rather than space (and time) pre-existing as a box in which physical processes unfold.

[@Deepakyadav-vp8xx]

Go to vector concept of the triangle

[@Hythloday71]

Whoa ! - Blow my mind ! - Riemann espouses physics over mathematics ? Are you sure this isn't a simplification ? I followed the geometry claims, but the almost, spiritual leaps to where you end up - well I didn't follow them, that is they seemed unjustified / spiritual ? Loved the vid mind, pretty mind blowing, perhaps it's something I need to acclimatise to.

[@PrometheanAction]

Thanks - the conclusion Riemann makes, after demonstrating that there are many different possible metrics for space, is that the basis upon which the *correct metric could be determined, cannot be discovered by a priori mathematical thinking, but only by physical experiment and discovery. (Think of what Einstein did, with general relativity: a physical principle-light motion-giving rise to a shape of space). That means that, at heart, the basis of space is not geometry, but physics. People are entire human beings: Riemann worked on physics and theology. What guided him in his indisputably correct work in mathematics?

[@JamesDavis-qk4hr]

I am with you, he jumps too far to be reasonable at the end.

[@angelmendez-rivera351]

Hythloday71 It definitely is a simplification

[@ParsevalMusic]

The question of my life: why Pi.. why such that irrational number linked to the most symmetrical shape... why that? thanks

[@jasonaross]

The circle has a basis for existence that lies outside of arithmetic. It is based on rotation, and also has the unique characteristic of enclosing the most area with the least circumference. The basics of arithmetic (+-×/) refer to lines, and the circle is curved, rather than straight. For more, see this video at 8 minutes: kzbin.info/www/bejne/jXTHnH16d9SFhdE

[@JB-yt4be]

Do you believe economics is a process in nature which can sustain exponential growth?

[@nal8503]

+Jordan butler Depends by what quantity you measure economic growth. If by the GDP, sure. Just keep "making up money" out of thin air. If you want that money to actually have applicable value though, you definitely need sufficient funding in science and education (engineering etc. included), as these are the drivers of human knowledge and accomplishment, not policies that regulate the innovations of the creative minds. Mind you, I'm not saying that policies aren't required, but they need to empower, not enslave or imprison innovation. But in short, yes. We can most likely grow indefinitely if we keep looking for new things. Every new piece of knowledge is like a _new type_ of building block. And it's abundantly clear to even a playing child that any extra building block (even if they are the same) dramatically increases the potential structures that can be built.

[@jasonaross]

+Jordan butler Whether it is specifically exponential, human economics has the potential to sustain unlimited growth. Since we will never run out of things to discover, there will always be more principles in the future to implement and use to transform our relation to the universe and each other. Embracing that fact of economics makes it a beautiful science.

[@gravitywaves123]

This guy has a gripe with pure mathematician.

[@samk8890]

nice vedio

[@johnchappell9232]

Economic development brings measure to your concept of infinite resource. So where is the discovery in measure? its only an extension of principle.

[@PrometheanAction]

Could you rephrase that? I don't quite understand your meaning...

[@ghiribizzi]

Is it itself? 0:33

[@PrometheanActionLive]

Could you please elaborate? I don't understand your question. - Jason

[@SamuelFoster666]

Omg, thank you for the revelation.

[@davidwilkie9551]

Mind is reiterative evolution of the same sequence of cycles at variable rates, all in one connection of minimum time. _____ The Temporal connection of something nothing interpenetrating existence is self-defining Superspin, a Superposition-singularity turned insideout Topologically by relative rates of time-duration, or Epicyclic superimposed Echo-location modulation, as distributed inflation-gravitation bubbles in QM-Time, ..and in Totality is the cause-effect Eternity Now /history of this Spinfoam Spacetime. It includes the Mind-body of Actual Intelligence, processing Quantum Information Image Calculations here in the Temporal Superposition Now. Time is the self-defining reason of motion-meaning for the substantiation of Actuality, the projected image of Physical Reality/duality of the Universal wave-package clock, Timing-spacing.

[@sicktoaster]

15:19 I thought you couldn't divide by 0.

[@LaroucheScience]

Thanks for the question. In this case, zero is the result of dividing by infinity. The measure of curvature is 1/Rr, where R and r are the radii of the extreme osculating circles. For the cylinder, one radius is the cylinder radius, while the other osculating circle, which is the straight line of the cylinder's side, has an infinite radius. So R·r = infinity, and 1/infinity is zero. How does that sound?

[@habiks]

You don't call it parallel line if they intersect in 1st place.. a true parallel line never intersects.. how will you know that the lines are parallel? It's when they won't intersect.. how hard can it be? you can continue talking about your thinking about lines in curved space that intersect now.

[@jasonaross]

Two lines that seem to be parallel when considered in a certain region of space may actually intersect, if the broader space they exist in is curved. The point of bringing this up is that the notion of parallel lines includes a concept (that they can be extended infinitely without meeting) that is actually a hypothesis about space, a hypothesis that people don't generally recognize they are making.

[@habiks]

Jason Ross if they meet in the broader space, you were wrong to call them parallel in the first place. We are talking about geometry and not curvature of spacetime (and even if space is some kind of sphere, then parallel lines don't exist in real space).

[@jasonaross]

Google+ SUCKS BALLS - the worst forced social network Whether parallel lines, depends on the geometry you are drawing them in. A flat space, in which such parallel lines can be drawn, is *one possible space, but is not necessary. This is Riemann's point. Answering the question means we must leave mathematics and go to physics.

[@ss11223344ful]

The mind would only have physical force if we have free will. It is not yet proven that we do. Our mind may be just an illusion caused by the physical processes in our brain.

[@polyhistorphilomath]

Problematic attack on Euclid. Suppose you measure the diameter of a sphere on the earth’s surface. Euclidean geometry differs not even the width of a single atom from the true length of a meter-wide sphere. Spoiler: Completely contradicted at 16:00. Eratosthenes is a genius-Euclid was dropped on his head. You win the internet!

[@quagapp]

But it is very good all the same.

[@DrunkenUFOPilot]

Illusion is possible only if there exists someone to observe it.

[@rahulv7530]

amazing,Thanks a lot!

[@frankdimeglio8216]

What is E=MC2 is consistent with TIME AND what is gravity. (TIME is thoroughly consistent with what is gravity ON/IN BALANCE.) WHAT IS E=MC2 is dimensionally consistent. TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE) !!! WHAT IS E=MC2 is taken directly from F=ma. Consider what is the man (AND THE EYE ON BALANCE) who IS standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). CLEAR water comes from what is THE EYE. INDEED, consider what is (essentially and necessarily) BALANCED BODILY/VISUAL EXPERIENCE !!! Lava IS orange, AND it is even blood red. The hottest flame is blue. The hottest lava is yellow. LOOK upwards, ON BALANCE, at what is the TRANSLUCENT AND BLUE sky !! The orange (AND setting) Sun IS the SAME SIZE as what is THE EYE !! NOW, consider what is the fully illuminated (AND setting/WHITE) MOON ON BALANCE. (BALANCE AND completeness go hand in hand.) WHAT IS E=MC2 is taken directly from F=ma, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. WHAT IS E=MC2 is taken directly from F=ma. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites ON BALANCE, as the stars AND PLANETS are POINTS in the night sky. Consider TIME AND time dilation ON BALANCE. c squared CLEARLY represents a dimension of SPACE ON BALANCE. WHAT IS GRAVITY is, ON BALANCE, an INTERACTION that cannot be shielded or blocked (ON BALANCE) !!! E=MC2 is consistent with/AS WHAT IS GRAVITY, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. Magnificent. Notice that the curvature or shape of said Moon matches that of what is THE EARTH/ground (that is, given what is a CLEAR horizon, of course.) The diameter of WHAT IS THE MOON IS about ONE QUARTER that of WHAT IS THE EARTH/ground ON BALANCE. Excellent !!! It ALL CLEARLY makes perfect sense ON BALANCE. Consistent WITH WHAT IS TIME, WHAT IS E=MC2 IS GRAVITY ON BALANCE. Finally, the average ocean tide is about 6 feet; AND said Sun manifests or forms at what is EYE LEVEL/BODY HEIGHT. The tidal range on the open ocean is about 3 feet. Notice, what is THE EARTH is ALSO BLUE ON BALANCE. Outstanding. Again, ON BALANCE, consider what is the fully illuminated (AND setting/WHITE) MOON ON BALANCE. The BULK DENSITY of WHAT IS THE MOON is comparable to that of (volcanic) basaltic lavas on THE EARTH/ground. The surface of WHAT IS THE MOON is chiefly composed of pumice. Excellent. ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE), AS the rotation of WHAT IS THE MOON matches the revolution; AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS WHAT IS E=MC2 is taken directly from F=ma. By Frank Martin DiMeglio

[@quagapp]

Riemann and this young man is postmodernist mathematician! He leaves out Gauss and others such as Lobachevksky But Euclid is valid according to a book I have if we are in space and "on" a sphere of infiinite radius - but I am interested in the confluence of maths and even art, philosophy etc - as a poet/artist etc In general I can understand WHY he calls Euclid a faker, but in saying this he "begs the question". I doubt he was a faker, he just thought (as we all do) as we are encultuated.

[@gravitywaves123]

I guess he is from the school of the intuitionist.

[@Adityarm.08]

why so much hatred for Euclid? and calling creativity a fundamental force sounds somewhat nuts. Mind could be traced to chemistry for all i know.

[@nal8503]

+Aditya Mishra It certainly could be, however, chemistry underlies rules as well. I don't see a problem with calling creativity a fundamental force as long as you _do not_ interpret it in a religious way. Humans are not really doing anything that nature hasn't been doing for all of time. Take what's there, smash it together, create new stuff. It's no wonder that we humans are creating in the same manner, after all, we are a product and part of nature.

[@Adityarm.08]

***** Maybe you are right.. But It really is one of those things that make you feel strange when you think about it :) I mean, what is creativity exactly, Activity in our brain which involves mixing thoughts to create something new, right? So it's something abstract that emerges out of underlying laws of chemistry. What you are saying is that this emergent phenomenon has similarities with the workings of the world at the most fundamental levels that we know of. Or is it just that we are finding patterns where they have no reason to be? I'm not really sure about it.

[@nal8503]

Aditya Mishra"So it's something abstract that emerges out of underlying laws of chemistry." Try it from the opposite approach. What is chemistry? I'm no chemist, so the jargon might be off, but essentially different, already existing, molecules or atoms can bond to create new things. If you think of heat, pressure and other things like that as "things", then you essentially smash those things together to get other types of stuff. Such ass, heat + water = vapor for example. Now let's go back a bit. What is water? Or any "chemical"? A collection of already existing particles which are thrown together in various ways to create larger structures, such as atoms which then create molecules. Basically, you can boil everything down to the same process, applied to different scales. Particles become atoms become molecules become organisms or structures (what is even the difference between living and dead "organisms"?) become larger animals. Thoughts, societies, creative endeavours of humans and anything else I can observe appear to function in the fundamentally same way. Basically, (metaphorically speaking), the universe is a child's toy box, which consists of an infinite amount of lego bricks which he puts together to make up new types of lego from what he already has. (I'm not religious and not trying to promote a supernatural god, I just like simple metaphors to bring points across.) "Or is it just that we are finding patterns where they have no reason to be?" As far as this question is concerned. I really don't know (anything). I sometimes like to think of the implications of the following assumption: What if the human brain isn't a "perception and thought creating machine" but instead a "filtering device that puts a noise gate on unwanted information"?

[@PoloMaldonadoM]

He hates Euclid because he's part of a group that mixes political ideology with authentic science, and looks for inventing conflict where there's none. Creativity is a fundamental force (easily seen its effects in the creation of civilizations) but the idea has nothing to do with his very precise earlier analysis which is very spot on. I don't believe the mind can be traced or reduced to chemistry, however discusing such idea belogns to another subject.

[@angelmendez-rivera351]

Nal The problem with calling something a fundamental force is that fundamental force already has a physical meaning. Also, calling creativity a fundamental force goes against the definition of what fundamental is, because creativity isn’t fundamental in nature by any means. As stated earlier, the brain can be reduced to chemistry, and chemistry can be reduced to physics. The idea behind something being fundamental is that it can’t be reduced into the foundations of something else like the brain can be. Creativity is a force, an awesome one, but fundamental is just a misnomer and saying that it is somehow okay is condoning the abuse of scientific language, which honestly should have never been condoned in history. The entire idea they try to promote with this is misleading at best

[@henryzorrilla5711]

Euclid was a faker? What is the physical interpretation of an infinite dimensional manifold? How does one reach any conclusions about Lp spaces through experimentation? Yes, understanding of a mathematical concept is usually aided by using the concept to model a physical process, but to say that logic has no place in mathematics is antithetical to the very definition of mathematics. I applaud your attempt to expose civilians to mathematics but please be careful of the ideas you maybe inspiring.

[@angelmendez-rivera351]

Henry Zorrilla This is exactly what I was saying. Truth in mathematics cares not for what the physical world is. In mathematics, we establish axioms, and if we can prove a theorem with the axioms, then the theorem is true, it does not matter how much it differs from what we can use or observe in the physical realm. This is why mathematicians are allowed to say that we can have negative squares, have quaternions, and extend fields into wheels to divide by zero, even though none of these are actually observable in the physical realm. Math itself is unobservable as every mathematical entity is abstract. Math is created in the language of logic. Physics is only there to tell us how to use the math practically, but physics is not what dictates 2+2=4. In fact, they’re fundamentally different. Truth is established axiomatically in math, but in science, truth is not a well defined concept. Instead, we’re concerned with data and observations empirically, and we want our hypotheses to match the data as much as it can, within reasonable intervals of confidence.

[@BobBogaert]

The last few minutes of the video degrade the whole thing from being a beautifully concise introduction to Riemann to being some wishy washy pseudo-philosophical treatise of which I can`t even see the goal. Is this supposed to make some 19y old kids think your political organisation consists of wise men and deep thinkers? I sure hope not. Apart from that, I have great respect for everyone putting the effort and resources in a free educational video, so I`ll pretend those 5 min ain`t there.

[@NihmbleTech]

Euclid's axioms are clearly defined to be valid within Euclidean space, and form a self consistent model. Anything above and beyond that are assumptions you're projecting on it with your own bias. The naive attack on Euclid is just silly. Calling Euclid a "faker" is laughable and undercuts all of the extremely useful and practical achievements that have emerged from his work. It also fails to recognize that the pioneering greats who laid the foundations for those to follow did so without much of the endowed knowledge you seemingly take for granted. Let's come back in 2000 years and see how much of your work and contribution to humanity compares to Euclid's and maybe you will have some shred of credibility.

[@locallyringedspace3190]

I agree that whole part was very distasteful

[@TaterGumfries]

Callin Euclid a faker (or did you mean fakir?) just stopped Tater in his tracks. If you ain't got the sense to put his work in context, then you don't deserve no more attention. You don't know enough to lecture on Riemann.

[@jasonaross]

Perhaps you could have listened to the rest of the video, which does precisely that-putting my objections to Euclid in context.

[@nextplanetgaming8749]

Apparently, in your world, it's normal to judge a thesis based upon its preamble. Thank goodness intellectuals don't live in your world.

[@PoloMaldonadoM]

You cannot escape from Euclid's definitions. You even demonstrated their validity at some point. In the moment you use "radius"...well, you are using Euclid's definitions.

[@jonathanbourret2968]

Apparently "Faker" is a trigger for some. Maybe you should have posited "Euclid had it backwards" (i.e. he had his physics match his math instead of the other way round).

[@jasonaross]

Polo -- the concept of a circle, and that it is made of locations an equal distance from a center, was not invented by Euclid. He did not invent geometry.